P. HISTORY OF ' AATHEMATICAL - School of Mathematics

TRIGONOMETRY
175
tan [(ct>, x)] =tan ct> sec x+tan X sec ct>. Let this function (ct>, X) be designated
ct>..lx, so that tan (ct>..lx) =tan ct> sec x+tan X sec ct>; let it be
designated (ct>Tx) when tan (ct>Tx) = tan ct> sec x-tan X sec ct>. Thus
when tan ct> is changed into i sin ct>, sec ct> into cos ct>, and cot ct> into
- i cosec ct>, ..1 must be changed into +, T, into -.
532. Inverse trigonometric functions.--Daniel Bernoulli was the
first to use a symbolism for inverse trigonometric functions. In 1729
he used "A S." to represent "arcsine."! Euler- in 1736 introduced
"A t" for "arctangent," in the definition: "expressio A t nobis
denotet arcum circuli, cuius tangens est t existente radio = 1."
Later, in the same publication, he expressed" the arcsine simply by
"A": "arcus cuius sinus est ~ existente toto sinu = 1 notetur per
a
A ~." To remove the ambiguity of using "A" for two different arc
a
functions, he introduced- the sign "A t" for "arctangent," saying,
"At. ~ est arcus circuli cuius tangens est ~ existente sinu toto = 1."
He used the sign"A t" in 1736 also in another place.' In 1737 Eulers
put down "A sin~" and explained this as meaning the arc of a unite
circle whose sine is ~. In 1744 he? uses "A tagt" for arcus, cujus tanc
gens= t. Lambert- fourteen years later says, "erit ~ = ~ arcus sinui b
m 0
respondens." Carl Soherffer'' in Vienna gives "arc. tang." or "arc.
tangent"; J. Lagrange" in the same year writes "arc. sin 1~a ." In
1 Daniel Bernoulli in Comment. acado sc, Petrop., Vol. II (1727; printed 172S),
p.304-42. Taken from G. Enestrom, Bibliotheca maihemaiica (3d ser)., Vol. XIV,
p. 78. See also Vol. VI (1905), p. 319-21.
2 L. Euler, Mechanica sive motus scientia (Petropoli, 1736), Vol. I, p. 184-86.
B L. Euler, op. cii., Vol. II, p. 138. • L. Euler, op, cit., Vol. II, p. 303.
• L. Euler, Comment. acado sc, Petrop., Vol. VIII (1736; printed in 1741),
p. 84,85.
• L. Euler in Commentarii academiae Petropoluaruie ad annum 1737, Vol. IX,
p. 209; M. Cantor, op. cii., Vol. III (2d ed.), p. .560.
t Nova acta eruditorum (1744), p. 325; Cantor, op. cit., Vol. III (2d ed.), p. 560.
B Acta Helvetica Physico-Maih.-Botan., Vol. III (Basel, 1758), p. 141.
9 Carolo ScherfIer, Institutionum analyticarum pars secunda (Vienna, 1772),
p, 144,200.
10 J. Lagrange in Nouoeauarmemoires de l'acad6nie r, d. sciences et belles-lettres,
annee 1772 (Berlin, 1774), p. 277.